Designing a Search Robot to Find a Beacon

New York State Common Core Math Geometry, Module 4, Lesson 4

Worksheets for Geometry

Student Outcomes

Given a segment in the coordinate plane, students find the segments obtained by rotating the given segment by 90° counterclockwise and clockwise about one endpoint.

Designing a Search Robot to Find a Beacon

Classwork

Opening Exercise

Write the equation of the line that satisfies the following conditions:
a. Has a slope of 𝑚 = −1/4 and passes through the point (0,−5).
b. Passes through the points (1,3) and (−2,−1).

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Exploratory Challenge

A search robot is sweeping through a flat plane in search of the homing beacon that is admitting a signal. (A homing beacon is a tracking device that sends out signals to identify the location). Programmers have set up a coordinate system so that their location is the origin, the positive 𝑥-axis is in the direction of east, and the positive 𝑦-axis is in the direction of north. The robot is currently 600 units south of the programmers’ location and is moving in an approximate northeast direction along the line 𝑦 = 3𝑥 − 600.

Along this line, the robot hears the loudest “ping” at the point (400, 600). It detects this ping coming from approximately a southeast direction. The programmers have the robot return to the point (400, 600). What is the equation of the path the robot should take from here to reach the beacon?

Begin by sketching the location of the programmers and the path traveled by the robot on graph paper; then, shade the general direction the ping is coming from.

Example

The line segment connecting (3,7) to (10,1) is rotated clockwise 90° about the point (3,7).
a. Plot the segment.

b. Where will the rotated endpoint land?

c. Now rotate the original segment 90° counterclockwise. Before using a sketch, predict the coordinates of the rotated endpoint using what you know about the perpendicular slope of the rotated segment.

Exercise

The point (𝑎, 𝑏) is labeled below:
a. Using 𝑎 and 𝑏, describe the location of (𝑎, 𝑏) after a 90° counterclockwise rotation about the origin. Draw a rough sketch to justify your answer.
b. If the rotation was clockwise about the origin, what is the rotated location of (𝑎, 𝑏) in terms of 𝑎 and 𝑏?

Draw
a rough sketch to justify your answer.
c. What is the slope of the line through the origin and (𝑎, 𝑏)? What is the slope of the perpendicular line through the origin?
d. What do you notice about the relationship between the slope of the line through the origin and (𝑎, 𝑏) and the slope of the perpendicular line?

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